though nowadays they look at things like "supercore service inflation" and core ex shelter and who knows what other series.
First the regular headline CPI number, and the regular month-by-month increases (rolling 3 months stuff comes later) :

After a frightening late summer jump, a nice downward trend to 0% in October (actually 0.04% which annualizes to 0.54%)
Now the CORE CPI number that the Fed is more interested in (though actually its the CORE PCE inflation gauge that historically has been their #1 gauge of underlying inflation trend to project FUTURE inflation)

After a bump-up in late summer, a nice move downward in October back to the June-July numbers.
CPI Rolling 3 month average thru October 2023:
For better accuracy, I calculate the rolling 3 months average using the actual index numbers, not the one digit monthly change numbers
A rolling 3 month helps smooth out month-to-month volatility, and since 3 months is an average of 3 data points, it is less likely to be dismissed as a "one off" like a single month's increase could be.

The big August & September jump hurt the 3 month average. As August and September fall out of the 3 month average in the coming one and two months, there should be a dramatic improvement.
CORE CPI Rolling 3 month average thru October 2023:

It has finally come down to about the 3% range. As August and September fall out of the 3 month average in the coming months, this may well improve considerably.
The next Fed rate-setting meeting is December 12-13.
BLS CPI press release: https://www.bls.gov/news.release/cpi.nr0.htm
Various series:
CPI: https://data.bls.gov/timeseries/CUSR0000SA0&output_view=pct_1mth
CORE CPI: http://data.bls.gov/timeseries/CUSR0000SA0L1E&output_view=pct_1mth
For all BLS timeseries data, one can see the index values and other periods like rolling 3 month, rolling 6 month, rolling 12 months by clicking "More Formatting Options" on the upper right and then on the page that shows up, check the various checkboxes
REAL AVERAGE HOURLY EARNINGS of production and non-supervisory workers https://data.bls.gov/timeseries/CES0500000032 ,
. . . private workers: https://data.bls.gov/timeseries/CES0500000013
CPI excluding shelter - https://data.bls.gov/timeseries/CUUR0000SA0L2
. . . FRED: https://fred.stlouisfed.org/series/CUUR0000SA0L2
. . . Table 3 has CPI ex shelter, as well as Core ex shelter https://www.bls.gov/news.release/cpi.t03.htm
Rent (SA) https://data.bls.gov/timeseries/CUSR0000SEHA
Fred: (SA) Rent of Primary Residence in U.S. City Average https://fred.stlouisfed.org/series/CUSR0000SEHA
(NSA) https://fred.stlouisfed.org/series/CUSR0000SEHA
SA = Seasonally Adjusted, NSA = Not Seasonally Adjusted